Efficient road coordinates transformations library

ABSTRACT

A system and method operate an autonomous vehicle. A sensor senses a road and an object. A processor determines, in a Cartesian reference frame, a representation of the road and a source point representative of the object, samples a first waypoint and a second waypoint from the representation of the road, determines a linear projection of the source point to a line connecting the first waypoint and the second waypoint, determines a first estimate of a longitudinal component of the source point in a road-based reference frame based on the linear projection, the first estimate being on a curve representing the road between the first waypoint and the second waypoint, determines a second estimate of the longitudinal component from the first estimate, determines a coordinate of the source point in the road-based reference frame from the second estimate and operates the vehicle with respect to the object using the coordinate.

INTRODUCTION

The subject disclosure relates to autonomous vehicles and, inparticular, to a system and method for efficient projection of objectsonto a road-based reference frame.

A partially or fully autonomous vehicle tracks objects in itsenvironment and performs navigational maneuvers with respect to theseobjects. The objects are sensed using a sensor at the vehicle and mappedto a Cartesian coordinate system. However, for computational efficiency,it is desirable to perform calculations in a road-centered coordinatesystem (also known as a Frenet space) that moves along a road, off-roadpath, curve or path-like construction. Transforming from the Cartesiancoordinate system to the Frenet space can become computationallyexpensive if applied at scale. Accordingly, it is desirable to provide acomputationally efficient system and method for transforming an object'scoordinates from a Cartesian coordinate system to a road-centeredcoordinate system.

SUMMARY

In one exemplary embodiment, a method of operating an autonomous vehicleis disclosed. A representation of a road and a source pointrepresentative of an object is determined in a Cartesian referenceframe. A first waypoint and a second waypoint are sampled from therepresentation of the road. A linear projection of the source point to alinear segment connecting the first waypoint and the second waypoint isdetermined. A first estimate of a longitudinal component of the sourcepoint is determined in a road-based reference frame based on the linearprojection, the first estimate being located on a curve representing theroad and connecting the first waypoint and the second waypoint. A secondestimate of the longitudinal component of the source point is determinedin the road-based reference frame from the first estimate. A coordinateof the source point in the road-based reference frame is determined fromthe second estimate of the longitudinal component. The vehicle isoperated with respect to the object using the coordinate of the sourcepoint in the road-based reference frame.

In addition to one or more of the features described herein, the secondestimate of the longitudinal component is determined using a circulararc approximation to the curve. The second estimate is realized in theCartesian reference frame by interpolating a closest waypoint to thesecond estimate using a Taylor series approximation and Frenet-Serretformulas. At least the first waypoint and the second waypoint aregrouped into a plurality of waypoint clusters and a waypoint cluster isselected from the plurality of waypoint clusters based on a distancefrom the waypoint cluster to the source point. A source cluster isformed that includes the source point and the waypoint cluster isselected based on the distance between the source cluster and thewaypoint cluster. The method further includes determining a lateralcomponent of the source point using the second estimate of thelongitudinal component. In an embodiment, the representation of the roadis a spline of polynomials.

In another exemplary embodiment, a system for operating an autonomousvehicle is disclosed. The system includes a sensor and a processor. Thesensor senses a road and an object. The processor is configured todetermine, in a Cartesian reference frame, a representation of the roadand a source point representative of the object, sample a first waypointand a second waypoint from the representation of the road, determine alinear projection of the source point to a line connecting the firstwaypoint and the second waypoint, determine a first estimate of alongitudinal component of the source point in a road-based referenceframe based on the linear projection, the first estimate located on acurve representing the road and connecting the first waypoint and thesecond waypoint, determine a second estimate of the longitudinalcomponent of the source point in the road-based reference frame from thefirst estimate, determine a coordinate of the source point in theroad-based reference frame from the second estimate of the longitudinalcomponent, and operate the vehicle with respect to the object using thecoordinate of the source point in the road-based reference frame.

In addition to one or more of the features described herein, theprocessor is further configured to determine the second estimate using acircular arc approximation to the curve. The processor is furtherconfigured to realize the second estimate in the Cartesian referenceframe by interpolating a closest waypoint to the second estimate using aTaylor series approximation and Frenet-Serret formulas. The processor isfurther configured to group at least the first waypoint and the secondwaypoint into a plurality of waypoint clusters and select a waypointcluster from the plurality of waypoint clusters based on a distance fromthe waypoint cluster to the source point. The processor is furtherconfigured to form a source cluster that includes the source point andselect the waypoint cluster based on the distance between the sourcecluster and the waypoint cluster. The processor is further configured todetermine a lateral component of the source point using the secondestimate of the longitudinal component. In an embodiment, therepresentation of the road is a spline of polynomials.

In yet another exemplary embodiment, an autonomous vehicle is disclosed.The autonomous vehicle includes a sensor and a processor. The sensorsenses a road and an object. The processor is configured to determine,in a Cartesian reference frame, a representation of the road and asource point representative of the object, sample a first waypoint and asecond waypoint from the representation of the road, determine a linearprojection of the source point to a line connecting the first waypointand the second waypoint, determine a first estimate of a longitudinalcomponent of the source point in a road-based reference frame based onthe linear projection, the first estimate located on a curverepresenting the road and connecting the first waypoint and the secondwaypoint, determine a second estimate of the longitudinal component ofthe source point in the road-based reference frame from the firstestimate, determine a coordinate of the source point in the road-basedreference frame from the second estimate of the longitudinal component,and operate the vehicle with respect to the object using the coordinateof the source point in the road-based reference frame.

In addition to one or more of the features described herein, theprocessor is further configured to determine the second estimate using acircular arc approximation to the curve and realize the second estimatein the Cartesian frame by interpolating the closest waypoint to thesecond estimate using a Taylor series approximation and Frenet-Serretformulas. The processor is further configured to group at least thefirst waypoint and the second waypoint into a plurality of waypointclusters, from a source cluster that includes the source point, andselect a waypoint cluster from the plurality of waypoint clusters basedon a distance between the source cluster and the waypoint cluster. Theprocessor is further configured to determine a lateral component of thesource point using the second estimate of the longitudinal component. Inan embodiment, the representation of the road is a spline ofpolynomials. In an embodiment, the representation of the road is a lanecenter of the road.

The above features and advantages, and other features and advantages ofthe disclosure are readily apparent from the following detaileddescription when taken in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

Other features, advantages and details appear, by way of example only,in the following detailed description, the detailed descriptionreferring to the drawings in which:

FIG. 1 shows a vehicle in accordance with an exemplary embodiment;

FIG. 2 illustrates a transformation process between a Cartesianreference frame and a road-centered reference frame;

FIG. 3 shows a mathematical representation of a road;

FIG. 4 shows a clustering method for associating a source point withnearest neighboring waypoints;

FIG. 5 illustrates a method of obtaining an estimate of a longitudinalcomponent of a source point in a road-centered reference frame;

FIG. 6 shows a flowchart illustrating a method for determining alongitudinal component for a source point in a road-centered referenceframe; and

FIG. 7 shows a flowchart for transforming from the road-centeredreference frame back to the Cartesian reference frame.

DETAILED DESCRIPTION

The following description is merely exemplary in nature and is notintended to limit the present disclosure, its application or uses. Itshould be understood that throughout the drawings, correspondingreference numerals indicate like or corresponding parts and features.

In accordance with an exemplary embodiment, FIG. 1 shows a vehicle 10.In an exemplary embodiment, the vehicle 10 is a semi-autonomous orautonomous vehicle. In various embodiments, the vehicle 10 includes atleast one driver assistance system for both steering andacceleration/deceleration using information about the drivingenvironment, such as cruise control and lane-centering. While the drivercan be disengaged from physically operating the vehicle 10 by having hisor her hands off the steering wheel and foot off the pedal at the sametime, the driver must be ready to take control of the vehicle.

In general, a trajectory planning system 100 determines a trajectoryplan for automated driving of the vehicle 10. The vehicle 10 generallyincludes a chassis 12, a body 14, front wheels 16, and rear wheels 18.The body 14 is arranged on the chassis 12 and substantially enclosescomponents of the vehicle 10. The body 14 and the chassis 12 may jointlyform a frame. The front wheels 16 and rear wheels 18 are eachrotationally coupled to the chassis 12 near respective corners of thebody 14.

As shown, the vehicle 10 generally includes a propulsion system 20, atransmission system 22, a steering system 24, a brake system 26, asensor system 28, an actuator system 30, at least one data storagedevice 32, at least one controller 34, and a communication system 36.The propulsion system 20 may, in various embodiments, include aninternal combustion engine, an electric machine such as a tractionmotor, and/or a fuel cell propulsion system. The transmission system 22is configured to transmit power from the propulsion system 20 to thefront wheels 16 and rear wheels 18 according to selectable speed ratios.According to various embodiments, the transmission system 22 may includea step-ratio automatic transmission, a continuously variabletransmission, or other appropriate transmission. The brake system 26 isconfigured to provide braking torque to the front wheels 16 and rearwheels 18. The brake system 26 may, in various embodiments, includefriction brakes, brake by wire, a regenerative braking system such as anelectric machine, and/or other appropriate braking systems. The steeringsystem 24 influences a position of the front wheels 16 and rear wheels18. While depicted as including a steering wheel for illustrativepurposes, in some embodiments contemplated within the scope of thepresent disclosure, the steering system 24 may not include a steeringwheel.

The sensor system 28 includes one or more sensing devices 40 a-40 n thatsense observable conditions of the exterior environment and/or theinterior environment of the vehicle 10, including an object 50. Theobject 50 can be other road users, road elements such as a lane centeror lane edge, or other objects. The sensing devices 40 a-40 n caninclude, but are not limited to, radars, lidars, global positioningsystems, optical cameras, thermal cameras, ultrasonic sensors, and/orother sensors for observing and measuring parameters of the exteriorenvironment. The sensing devices 40 a-40 n may further include brakesensors, steering angle sensors, wheel speed sensors, etc. for observingand measuring in-vehicle parameters of the vehicle. The cameras caninclude two or more digital cameras spaced at a selected distance fromeach other, in which the two or more digital cameras are used to obtainstereoscopic images of the surrounding environment in order to obtain athree-dimensional image. The actuator system 30 includes one or moreactuator devices 42 a-42 n that control one or more vehicle featuressuch as, but not limited to, the propulsion system 20, the transmissionsystem 22, the steering system 24, and the brake system 26. In variousembodiments, the vehicle 10 can further include interior and/or exteriorfeatures such as, but are not limited to, doors, a trunk, and cabinfeatures such as air, music, lighting, etc. (not numbered).

The at least one controller 34 includes a processor 44 and a computerreadable storage device or media 46. The processor 44 can be any custommade or commercially available processor, a central processing unit(CPU), a graphics processing unit (GPU), an auxiliary processor amongseveral processors associated with the at least one controller 34, asemiconductor-based microprocessor (in the form of a microchip or chipset), a macroprocessor, any combination thereof, or generally any devicefor executing instructions. The computer readable storage device ormedia 46 may include volatile and nonvolatile storage in read-onlymemory (ROM), random-access memory (RAM), and keep-alive memory (KAM),for example. KAM is a persistent or non-volatile memory that may be usedto store various operating variables while the processor 44 is powereddown. The computer-readable storage device or media 46 may beimplemented using any of a number of known memory devices such as PROMs(programmable read-only memory), EPROMs (electrically PROM), EEPROMs(electrically erasable PROM), flash memory, or any other electric,magnetic, optical, or combination memory devices capable of storingdata, some of which represent executable instructions, used by the atleast one controller 34 in controlling the vehicle 10.

The instructions may include one or more separate programs, each ofwhich comprises an ordered listing of executable instructions forimplementing logical functions. The instructions, when executed by theprocessor 44, receive and process signals from the sensor system 28,perform logic, calculations, methods and/or algorithms for automaticallycontrolling the components of the vehicle 10, and generate controlsignals to the actuator system 30 to automatically control thecomponents of the vehicle 10 based on the logic, calculations, methods,and/or algorithms. Although only one controller is shown in FIG. 1 ,embodiments of the vehicle 10 can include any number of controllers thatcommunicate over any suitable communication medium or a combination ofcommunication mediums and that cooperate to process the sensor signals,perform logic, calculations, methods, and/or algorithms, and generatecontrol signals to automatically control features of the vehicle 10.

The communication system 36 is configured to wirelessly communicateinformation to and from other entities 48, such as but not limited to,other vehicles (“V2V” communication) infrastructure (“V2I”communication), remote systems, and/or personal devices. In an exemplaryembodiment, the communication system 36 is a wireless communicationsystem configured to communicate via a wireless local area network(WLAN) using IEEE 802.11 standards or by using cellular datacommunication. However, additional or alternate communication methods,such as a dedicated short-range communications (DSRC) channel, are alsoconsidered within the scope of the present disclosure. DSRC channelsrefer to one-way or two-way short-range to medium-range wirelesscommunication channels specifically designed for automotive use and acorresponding set of protocols and standards.

While the vehicle is disclosed herein as a fully autonomous or partiallyautonomous vehicle, this is not meant to be limitation of the invention.In other embodiments, the vehicle can be any type of partiallyautonomous or fully autonomous vehicle, such as a motorcycle, scooter,or mobile robot, for example.

FIG. 2 illustrates a transformation process 200 between a Cartesianreference frame 202 and a road-centered reference frame 204. TheCartesian reference frame 202 is stationary with respect to a road 205,while the road-centered reference frame 204 moves along the road. In anembodiment, the road-centered reference frame 204 moves along a lanecenter 210 of the road 205. In the road-centered reference frame 204,the road 205 can appear as a straight line. The road 205 can include acurved section as shown in the Cartesian reference frame 202. Thevehicle 10 senses various source points associated with objects in theCartesian reference frame 202. The various source points can include thepositions or coordinates of the vehicle 10 itself, other road users, thelane center 210 of the road 205, other objects, etc. Each source pointmay be parametrized within the Cartesian reference frame 202 using 6degrees of freedom, including the x, y, yaw, velocity, acceleration andcurvature (κ) for the source point. The same source point may beparametrized within the road-centered reference frame 204 by alongitudinal position (s) of the source point, a lateral position (d) ofthe source point as well as their relative time-derivatives (e.g.velocities and accelerations) along those two axes. Thus, the sourcepoint may be parameterized in the road-centered reference frame by(s,{dot over (s)},{umlaut over (s)},d,{dot over (d)},{umlaut over (d)}).To navigate the vehicle 10, the Cartesian coordinates of the sourcepoints are transformed into the road-centered reference frame 204.Computations for determining a trajectory for the vehicle 10 are thenperformed in the road-centered reference frame 204. The trajectory istransformed back from the road-centered reference frame 204 into theCartesian reference frame 202 in order to be tracked by vehicle 10.

FIG. 3 shows a mathematical representation of the road 205 (i.e., of alane center 210 of the road 205). In various embodiments, the sensingdevices 40 a-40 n sense the lane center 210 and the processor 44represents the lane center 210 as a polynomial function. In otherembodiments, the representation of the road 205 can be a precomputedquantity that is stored in a map database. The precomputedrepresentation can be, for example, transmitted to the vehicle from asatellite, street camera or other external device. In particular, thelane center 210 is represented as a spline of polynomial functions. Forthe illustrative spline shown in FIG. 3 , a first polynomial functionrepresents a section 302 between rear waypoint 310 a and middle waypoint310 d. A second polynomial function represents a section 304 betweenmiddle waypoint 310 d and front waypoint 310 f The first polynomialfunction and second polynomial function are continuous and continuouslydifferentiable at the middle waypoint 310 d.

The processor 44 samples each polynomial function in order to generate aset of waypoints. Sampling the first polynomial function generateswaypoints 310 b and 310 c. Sampling the second polynomial functiongenerates waypoint 301 e. The processor 44 samples the polynomialfunctions uniformly so that the distance between waypoints along apolynomial function is the same. Each waypoint 310 a-310 f has anassociated set of waypoint statistics, which includes a position (x,y)of the waypoint, the two-dimensional tangent and Normal Vectors ({rightarrow over (T)}, {right arrow over (N)}) of the waypoint, the localcurvature and first derivative of the local curvature (κ, κ′) at thewaypoint. The waypoints give an approximation of the road-centeredreference frame.

Sampling of the waypoints from the polynomial can be performed eitheronline (at the vehicle) or offline (away from the vehicle). In onlinesampling, the polynomial functions representing the road are stored in astorage database. The online processor queries the database for aselected polynomial function and samples the polynomial to obtain thewaypoints and their waypoint statistics. In offline sampling, thesampling of waypoints from the polynomial is performed offline. Theonline processor then queries the database for the waypoints directly.Online sampling is generally more storage-effect but requires moreonline computations. Offline sampling is generally morestorage-inefficient but requires less online computation.

FIG. 4 shows a clustering method for associating a source point withnearest neighboring waypoints on the illustrative spline of FIG. 3 . Thesource points can include the vehicle 10, other road users, staticobjects, destination points, or any other object relevant to trajectoryplanning for the vehicle. The clustering method groups source pointsinto one or more source clusters and groups waypoints into one or morewaypoint clusters. Distances between a selected source cluster and theone or more waypoint clusters can be determined. For the selected sourcecluster, the processor 44 selects one more waypoint clusters having ashortest or minimum distance to the selected source cluster as beingrelevant to the selected source cluster. By determining the location ofthe relevant waypoints in the manner illustrated in FIG. 4 , the need tomatch the source points to all waypoints in order to find the nearestneighbor waypoints is eliminated.

As shown in FIG. 4 for the illustrative spline, waypoints 310 a and 310b have been grouped into a first waypoint cluster 402 a, waypoints 310 cand 310 d have been grouped into a second waypoint cluster 402 b, andwaypoints 310 e and 310 f have been grouped into a third waypointcluster 402 c. Source points 404 have been grouped into a first sourcecluster 408 a, and sources 406 have been grouped into a second sourcecluster 408 b. Distances are determined between the first source cluster408 a and waypoint clusters 402 a, 402 b and 402 c. The smallestinter-cluster distances are between the first source cluster 408 a andthe first waypoint cluster 402 a and between the first source cluster408 a and the second waypoint cluster 402 b. Therefore, the firstwaypoint cluster 402 a and second waypoint cluster 402 b are selected asrelevant to source points 404 of the first source cluster 408 a.Similarly, comparison of distances associates the second source cluster408 b with the second waypoint cluster 402 b and third waypoint cluster402 c. As an example, a KD-Tree can be used to group the waypoints intoclusters, such that querying for a nearest neighboring waypoint to aselected source point in space (e.g. source point or cluster centroid)is done efficiently and saves computation.

FIG. 5 illustrates a method of obtaining an estimate of a longitudinalcomponent of a source point in a road-centered reference frame. A sourcepoint 502 is shown associated with waypoints 310 d and 310 e. A region506 including waypoints 310 d, 310 e and source point 502 is shown inclose-up to illustrate a process of projecting the source point from aCartesian reference frame into a road-centered reference frame.

Referring to the close-up of the region 506, a linear segment 512connects a first waypoint (i.e., waypoint 310 d) and a second waypoint(i.e., waypoint 310 e) of the spline. Curve 514 represents thecontinuous realization of the road's curve (i.e. the relevant segment ofthe polynomial spline) with which waypoints 310 d and 310 e coincide.

In a first step, a projection line 518 is drawn perpendicular to linearsegment 512 and passing through the source point 502. An intersectionbetween the projection line 518 and the linear segment 512 generates alinear projection 516 of the source point 502 on linear segment 512. Ina second step, the linear projection 516 is used to determine a firstestimate 520 of a longitudinal component of the source point 502 in theroad-based reference frame. The first estimate 520 is also referred toherein as an S-value estimate (s^([0])). The first estimate 520 islocated on curve 514.

In one embodiment, a distance along linear segment 512 between thelinear projection 516 and the waypoint 310 d is determined and anequivalent distance from the waypoint 310 d along curve 514 isdetermined to find the location of the first estimate 520. Aninterpolation is then performed using the first estimate 520 todetermine a second estimate 522 of the longitudinal component of thesource point 502 in the road-based reference frame. The second estimate522 is also referred to herein as a fine-grained estimate s* and islocated on curve 514.

The interpolation involves a Taylor expansion of Frenet-Serret formulasfor a particle moving along a curve. By assuming constant-radius forcurve 514 at the locality of the first estimate 520, a unit normalvector 526 (i.e., unit vector {right arrow over (N^([0]))}) isdetermined at the location of the first estimate 520. The realization ofthe first estimate 520 on curve 514 is represented continuously by apolynomial. Since the location of the first estimate 520 is performedonline, the processor has access only to waypoints. Curve 514 istherefore a reconstruction of the polynomial from the waypoint 310 d andwaypoint 310 e. The realization of the first estimate 520 on curve 514gives the radius 524 of a tangent circle to curve 514 at the location ofthe first estimate 520. This tangent circle is represented by its originO (point 510) and has a radius r. A vector relation shown in Eq. (1)relates the source point 502, origin 510 and first estimate 520:

$\begin{matrix}{\overset{\rightarrow}{Op} = {{r \cdot \overset{\rightarrow}{N^{\lbrack 0\rbrack}}} + \overset{\rightarrow}{p⁢{\alpha ⁡\left( s^{\lbrack 0\rbrack} \right)}}}} & {{Eq}.(1)}\end{matrix}$where s^([0]) is a distance travelled along the curve 514 from waypoint310 d and α(s^([0])) is the coordinates of s^([0]) in the Cartesianreference frame (i.e., x, y, {right arrow over (T)}, {right arrow over(N)} κ, κ′) at the location of s^([0]) along the curve 514 th. An angleβ between the unit normal vector 526 and the origin-source vector 528 isdetermined using the law of cosines as shown in Eq. (2):

$\begin{matrix}{{\cos(\beta)} = \left. {{dot}{\left( {\overset{\rightarrow}{N^{\lbrack 0\rbrack}} \cdot \overset{\rightarrow}{Op}} \right)/}} \middle| \overset{\rightarrow}{Op} \right|} & {{Eq}.(2)}\end{matrix}$Once the angle β has been determined from Eq. (2), the second estimate522 is determined using Eq. (3):s*=s ^([0]) +β·r  Eq. (3)

Once the second estimate 522 (i.e., s*) has been determined, arealization α(s^([0])) can be determined by traveling along curve 514 tolongitude s* by distance subtending angle β. A lateral component of thesource point 502 in the road-centered reference frame can be determinedusing the realization of the second estimate α(s*), where

$\begin{matrix}{\overset{\rightarrow}{{\alpha\left( s^{*} \right)}p} = {\overset{\rightarrow}{Op} - \overset{\rightarrow}{O{\alpha\left( s^{*} \right)}}}} & {{Eq}.(4)}\end{matrix}$

From these components, velocities and accelerations of the source point502 can be determined in the road-centered reference frame and atrajectory for the autonomous vehicle can be calculated, thus giving thecoordinates (s,{dot over (s)},{umlaut over (s)},d,{dot over (d)},{umlautover (d)}) in the road-centered reference frame.

FIG. 6 shows a flowchart 600 illustrating a method for determining alongitudinal component for a source point in a road-centered referenceframe. In box 602, Cartesian coordinates for the source point areobtained in a Cartesian reference frame as well as a plurality ofwaypoints of the road-based reference frame. In box 604, a set ofrelevant waypoints to the source point are determined. In variousembodiments, the set of relevant waypoints is determined by formingwaypoint clusters and a source cluster including the source point anddetermining which waypoint clusters are closest to the source cluster.In box 606, the source point is projected onto a line connecting a firstwaypoint and a second waypoint adjacent the first waypoint and from thesame waypoint cluster. The projection of the source point onto the linegenerates a linear projection of the source point.

In box 608, a first estimate of a longitudinal component of the sourcepoint in the road-centered reference frame is determined on thepolynomial spline segment connecting the first waypoint and secondwaypoint using the linear projection. In box 610, a second estimate ofthe longitudinal component of the source point in the road-centeredreference frame is determined from an interpolation of the firstestimate along a constant radius-arc approximation at the locality ofthe first estimate. Once the second estimate is determined, a lateralposition of the source point is determined based on the second estimateand a trajectory for the autonomous vehicle can be calculated in theroad-centered reference frame based on the source point.

FIG. 7 shows a flowchart 700 for transforming from the road-centeredreference frame back to the Cartesian reference frame. In box 702, theroad-centered coordinates for a selected point is determined. In box704, a projection of the selected point that lies on the spline ofreference is determined. The projection has the same longitudinalcoordinate as the selected point, with the lateral coordinate equal tozero). The cartesian realization of the projection is determined byeither sampling the continuous spline representation at the longitudecoordinate, or by finding the waypoint closest to the projection andinterpolating using a Taylor series approximation and Frenet-Serretformulas. The closest waypoint can be determined by rounding thelongitude coordinate of the projection to the closest value according tothe polynomial functions uniform sampling frequency discussed hereinwith respect to FIG. 3 . In box 706, the coordinates of the selectedpoint are transformed into the Cartesian reference frame using thestatistics (location, yaw, curvature and its first derivative) of thereference spline at the selected point.

While the above disclosure has been described with reference toexemplary embodiments, it will be understood by those skilled in the artthat various changes may be made and equivalents may be substituted forelements thereof without departing from its scope. In addition, manymodifications may be made to adapt a particular situation or material tothe teachings of the disclosure without departing from the essentialscope thereof. Therefore, it is intended that the present disclosure notbe limited to the particular embodiments disclosed, but will include allembodiments falling within the scope thereof.

What is claimed is:
 1. A method of operating an autonomous vehicle,comprising: determining, in a Cartesian reference frame, a polynomialrepresentation of a lane center of a road and a source pointrepresentative of an object; sampling a first waypoint and a secondwaypoint along the lane center from the polynomial representation of thelane center; determining a linear projection of the source point to alinear segment connecting the first waypoint and the second waypoint;determining a distance between the linear projection and the firstwaypoint along the linear segment; determining a first estimate of alongitudinal component of the source point in a road-based referenceframe based on the linear projection by placing the first estimate onthe polynomial representation of the lane center between the firstwaypoint and the second waypoint at an equivalent distance from thefirst way point; determining a second estimate of the longitudinalcomponent of the source point in the road-based reference frame from thefirst estimate, the second estimate located on the polynomialrepresentation; determining a coordinate of the source point in theroad-based reference frame from the second estimate of the longitudinalcomponent; and operating the vehicle with respect to the object usingthe coordinate of the source point in the road-based reference frame. 2.The method of claim 1, further comprising determining the secondestimate of the longitudinal component using a circular arcapproximation to the curve.
 3. The method of claim 1, further comprisingrealizing the second estimate in the Cartesian reference frame byinterpolating a closest waypoint to the second estimate using a Taylorseries approximation and Frenet-Serret formulas.
 4. The method of claim1, further comprising grouping at least the first waypoint and thesecond waypoint into a plurality of waypoint clusters and selecting awaypoint cluster from the plurality of waypoint clusters based on adistance from the waypoint cluster to the source point.
 5. The method ofclaim 4, further comprising forming a source cluster that includes thesource point and selecting the waypoint cluster based on the distancebetween the source cluster and the waypoint cluster.
 6. The method ofclaim 1, further comprising determining a lateral component of thesource point using the second estimate of the longitudinal component. 7.The method of claim 1, wherein the polynomial representation of the lanecenter of the road is a spline of polynomials.
 8. A system for operatingan autonomous vehicle, comprising: a sensor for sensing a lane center ofroad and an object; and a processor configured to: determine, in aCartesian reference frame, a polynomial representation of the lanecenter of the road and a source point representative of the object;sample a first waypoint and a second waypoint along the lane center fromthe polynomial representation of the lane center; determine a linearprojection of the source point to a line connecting the first waypointand the second waypoint; determine a distance between the linearprojection and the first waypoint along the linear segment; determine afirst estimate of a longitudinal component of the source point in aroad-based reference frame based on the linear projection by placing thefirst estimate on the polynomial representation of the lane centerbetween the first waypoint and the second waypoint at an equivalentdistance from the first way point; determine a second estimate of thelongitudinal component of the source point in the road-based referenceframe from the first estimate, the second estimate located on thepolynomial representation; determine a coordinate of the source point inthe road-based reference frame from the second estimate of thelongitudinal component; and operate the vehicle with respect to theobject using the coordinate of the source point in the road-basedreference frame.
 9. The system of claim 8, wherein the processor isfurther configured to determine the second estimate using a circular arcapproximation to the curve.
 10. The system of claim 8, wherein theprocessor is further configured to realize the second estimate in theCartesian reference frame by interpolating a closest waypoint to thesecond estimate using a Taylor series approximation and Frenet-Serretformulas.
 11. The system of claim 8, wherein the processor is furtherconfigured to group at least the first waypoint and the second waypointinto a plurality of waypoint clusters and select a waypoint cluster fromthe plurality of waypoint clusters based on a distance from the waypointcluster to the source point.
 12. The system of claim 11, wherein theprocessor is further configured to form a source cluster that includesthe source point and select the waypoint cluster based on the distancebetween the source cluster and the waypoint cluster.
 13. The system ofclaim 8, wherein the processor is further configured to determine alateral component of the source point using the second estimate of thelongitudinal component.
 14. The system of claim 8, wherein thepolynomial representation of the lane center of the road is a spline ofpolynomials.
 15. An autonomous vehicle, comprising: a sensor for sensinga lane center of road and an object; and a processor configured to:determine, in a Cartesian reference frame, a polynomial representationof the lane center of the road and a source point representative of theobject; sample a first waypoint and a second waypoint along the lanecenter from the polynomial representation of the lane center; determinea linear projection of the source point to a line connecting the firstwaypoint and the second waypoint; determine a distance between thelinear projection and the first waypoint along the linear segment;determine a first estimate of a longitudinal component of the sourcepoint in a road-based reference frame based on the linear projection byplacing the first estimate on the polynomial representation of the lanecenter between the first waypoint and the second waypoint at anequivalent distance from the first way point; determine a secondestimate of the longitudinal component of the source point in theroad-based reference frame from the first estimate, the second estimatelocated on the polynomial representation; determine a coordinate of thesource point in the road-based reference frame from the second estimateof the longitudinal component; and operate the vehicle with respect tothe object using the coordinate of the source point in the road-basedreference frame.
 16. The vehicle of claim 15, wherein the processor isfurther configured to determine the second estimate using a circular arcapproximation to the curve and realize the second estimate in theCartesian frame by interpolating the closest waypoint to the secondestimate using a Taylor series approximation and Frenet-Serret formulas.17. The vehicle of claim 15, wherein the processor is further configuredto group at least the first waypoint and the second waypoint into aplurality of waypoint clusters, from a source cluster that includes thesource point, and select a waypoint cluster from the plurality ofwaypoint clusters based on a distance between the source cluster and thewaypoint cluster.
 18. The vehicle of claim 15, wherein the processor isfurther configured to determine a lateral component of the source pointusing the second estimate of the longitudinal component.
 19. The vehicleof claim 15, wherein the polynomial representation of the lane center ofthe road is a spline of polynomials.